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Completing the square is a key method in algebra used to transform quadratic expressions into perfect square trinomials. This process helps in rewriting quadratic equations in a form that is easier to interpret. In this exercise, completing the square is applied to both the \(y\) and \(z\) variables of the given equation to recognize the geometric figure it represents.

To complete the square for a term of the form \(ax^2 + bx\):

• Halve the coefficient of the linear term \(b\).
• Square this result.
• Add and subtract this squared value to form a perfect square trinomial.

Let's break it down with the example provided in the exercise:

For \(y^2 - 2y\):

• Take \(\frac{-2}{2} = -1\).
• Square it: \((-1)^2 = 1\).
• Rewrite the expression: \(y^2 - 2y + 1 - 1\). This simplifies to \((y-1)^2 - 1\).

Similarly, for \(z^2 - 4z\), complete the square to get \((z-2)^2 - 4\).

Completing the square is essential for recognizing shapes like circles and spheres in geometry.

The standard form of a sphere's equation in three-dimensional space is critical for identifying spheres in geometry problems:

\((x-h)^2 + (y-k)^2 + (z-l)^2 = r^2\).

Here, \((h, k, l)\) represents the center of the sphere, and \(r\) is the radius. Simplifying the completed square format of a quadratic 3D equation to match this form will help us identify and describe the sphere's characteristics.

In the exercise, after completing the squares for \(y\) and \(z\), we rewrite the equation as:

• \(x^2 + (y-1)^2 + (z-2)^2 = 9\).

This can be recognized as the equation of a sphere because it matches the standard form with:

• Center \((0, 1, 2)\)
• Radius \(r = \sqrt{9} = 3\)

Understanding the standard form allows easy identification of the sphere's center and radius, which are crucial to geometrically describe it.

3D geometry involves studying shapes such as spheres, cylinders, and planes within three dimensions, defined by the axes \(x\), \(y\), and \(z\). A essential task is identifying and describing these shapes using algebraic equations.

A sphere is one of the simplest 3D forms, characterized by its uniform distance from a central point in all directions. The equation:

• \(x^2 + (y-1)^2 + (z-2)^2 = 9\)

from the exercise, illustrates this concept as it represents points that form a sphere.

In three-dimensional space, understanding the notion of center and radius is pivotal:

• The center \((0, 1, 2)\) determines the location.
• The radius \(3\) defines the size.

Grasping these components helps in visualizing and comprehending the spatial orientation and size of 3D shapes effectively. Being comfortable with equations in 3D geometry is fundamental to mastering more advanced geometric concepts.